Answer:
a) Probability distribution of the winnings is given in the table below
X ||| P(X)
35 | 0.0263
-1 | 0.9737
b) Mean = -$0.0532 = -$0.0532 to the nearest penny; that is, 5 pennies.
Standard deviation = $5.76 = 576 pennies to the nearest penny.
Explanation:
To construct the probability distribution, we first define the random variable X which represents the winnings of each spin
X can take on value of $35; that is, a win of $35 when the number of the slot the ball falls into matches the number you selected
And a value of -$1; that is, a loss of $1 if the number of the slot the ball falls into does not match the number you selected.
Note that there are 38 slots. And one can only select 1 slot at a time.
So, the probability that one wins when the slot that one selects is the right slot = (1/38) = 0.0263
Probability of losing = 1 - 0.0263 = 0.9737
So, the probability distribution is now given as
X ||| P(X)
35 | 0.0263
-1 | 0.9737
b) Mean and Standard deviation of the random variable X
Mean = Expected value.
Expected value = E(X) = Σ xᵢpᵢ
where xᵢ = each variable
pᵢ = probability of each variable
E(X) = (35)(0.0263) + (-1)(0.9737)
= 0.9205 - 0.9737 = -$0.0532 = -$0.05 to the nearest penny; thanks 5 pennies to the nearest penny.
Standard deviation = √(variance)
Variance = Var(X) = Σx²p − μ²
where μ = E(X) = -0.0532
Σx²p = (35²)(0.0263) + (-1)²(0.9737)
= 32.2175 + 0.9737 = 33.1912
Var(X) = 33.1912 - (-0.0532)² = 33.18836976
Standard deviation = √(variance)
= √(33.18836976) = $5.76; that is, 576 pennies.
Hope this Helps!!!